Spin-excitations of the quantum Hall ferromagnet of composite fermions

The spin excitations of a fractional quantum Hall system are evaluated within a bosonization approach. In a first step, we generalize Murthy and Shankar’s Hamiltonian theory of the fractional quantum Hall effect to the case of composite fermions with an extra discrete degree of freedom. Here, we mainly investigate the spin degrees of freedom, but the proposed formalism may be useful also in the study of bilayer quantum-Hall systems, where the layer index may formally be treated as an isospin. In a second step, we apply a bosonization scheme, recently developed for the study of the two-dimensional electron gas, to the interacting composite-fermion Hamiltonian. The dispersion of the bosons, which represent quasiparticle-quasihole excitations, is analytically evaluated for fractional quantum Hall systems at $\nu =1∕3$ and $\nu =1∕5$. The finite width of the two-dimensional electron gas is also taken into account explicitly. Furthermore, we consider the interacting bosonic model and calculate the lowest-energy state for two bosons. In addition to a continuum describing scattering states, we find a bound-state of two bosons. This state is interpreted as a pair excitation, which consists of a skyrmion of composite fermions and an antiskyrmion of composite fermions. The dispersion relation of the two-boson state is evaluated for $\nu =1∕3$ and $\nu =1∕5$. Finally, we show that our theory provides the microscopic basis for a phenomenological nonlinear $\sigma$ model for studying the skyrmion of composite fermions.

Figure 1

Schematic representation of the ground state of the 2DEG at ${\nu }^{*}=1$ (the quantum Hall ferromagnet of CFs).

Figure 2

Schematic representation of the neutral excitations of the quantum Hall ferromagnet of CFs at ${\nu }^{*}=1$. The bosonization method takes into account only spin-wave (SW) excitations which involve a spin reversal inside the lowest CF-LL. The excitations between different CF-LL, without a spin reversal [charge mode (CM)] and with a spin flip (SF) are neglected.

Figure 3

Dispersion relations of the bosons as a function of the momentum $\mathbf{q}$ with $\gamma =0$, in units of $e^2∕ϵl_B$, when (a) $\nu =1∕3$ and (b) $\nu =1∕5$. The solid line corresponds to Eq. (47), whereas the dashed and solid gray lines represent the results for the width parameters $\lambda =0.5$ and 1.0, respectively.

Figure 4

Dispersion relations of the two-boson state as a function of the total momentum $\mathbf{P}$, with $\gamma =0$, in units of $e^2∕ϵl_B$, when (a) $\nu =1∕3$ and (b) $1∕5$. The solid line corresponds to the lowest-energy state of two bosons, the dashed line is the energy of a well-separated CF quasiparticle-quasihole [Eq. (48)], and the shaded area represents the continuum of scattering states.